Manifolds which admit <span class="mathjax-formula">$\mathbf {R}^n$</span> actions

P UBLICATIONS MATHÉMATIQUES DE L ’I.H.É.S.

G ^ILLES C ^HATELET H AROLD R OSENBERG

Manifolds which admit R

actions

Publications mathématiques de l’I.H.É.S., tome 43 (1974), p. 245-260

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MANIFOLDS WHICH ADMIT R" ACTIONS

by G. GHATELET and H. ROSENBERG

INTRODUCTION

The purpose of this paper is to determine which TZ-manifolds admit smooth locally free actions of R""1. We shall restrict ourselves to compact connected orientable manifolds V" and locally free actions 9 ofR71"1 on V71 which are of class G2 and tangent to ^V", i.e. the components of W71 are orbits of 9. For n === 3, we know that V3 admits such an R2 action if and only if V^T^I or V3 is a bundle over S1 with fibre T2 [7].

Moreover, the topological type of such R2 actions has been completely determined [8].

We recall that the rank of V" is the largest integer k such that V71 admits a smooth locally free action of R^.

Now suppose that 9 is a locally free action ofR⁷¹"¹ on V⁷¹. We shall prove:

Theorem 1. — If 8V is not empty, then V is homeomorphic to T71"1 X I (here T denotes the torus of dimension z).

Theorem 2. — If W71 is empty and 9 has at least one compact orbit, then V71 is a bundle over S1 with fibre T71-1.

Theorem 3. — IfSV"' is empty and 9 has no compact orbits then V" is a bundle over a torus T^

with fibre a torus Tp-^.

Theorem 2 follows directly from Theorem i by cutting V71 along a compact orbit.

Theorem 3 depends upon an observation of Novikov [4], and independently Joubert:

suppose 9 acts on \rn with no compact orbits. By Sacksteder [9], all the orbits of 9 are

•pn-fc^^fc-i ^ gome k. Choose linearly independent vector fields X^, . . . , X ^ _ i tangent to the orbits of 9 such that all the integral curves ofX^, . . ., X^_^ are periodic, of period one. Then X^, . . ., X^_^ define a locally free action of ITn~k on V71 and the orbit space M is a smooth manifold of dimension k. Also M admits an action ofR^"1

with all the orbits R^"1. It follows that M is homeomorphic to T^, which proves Theorem 3 ([5] and [3]). Consequently, our main result is Theorem i. Here is how we proceed to prove Theorem i: by inductive arguments similar to those used in [7], we restrict ourselves to actions 9 with no compact orbits in the interior of V". We then

246 G . C H A T E L E T A N D H . R O S E N B E R G

remark that the foliation defined by the orbits of 9 is almost without holonomy, i.e. the noncompact leaves have no holonomy. With this, we construct collar neighborhoods U, of each component T, of^V, such that aU^T.uT^ with T,' transverse to the foliation.

We construct U, so that some linear field Y (tangent to the orbits of <p) is transverse to each T,'. We then prove the integral curves of Y go from T^ to T. hence define a homeomorphism of V71 to ^n~lxI.

i. Some Preliminaries.

( 1 . 1 ) Let yhe the filiation ofV defined by the orbits of 9. Then each noncompact leaf of y has zero holonomy.

Proof. — If T is a compact leaf of ^r, then the germ of ^ in a neighborhood of T is without holonomy outside ofT, provided T is an isolated compact leaf (page 13 of [8]).

This is ako true if T is an isolated compact leaf on one side in V and one considers the germ of ^ on this side. Now if 9 has no compact orbits then ^ is without holonomy and we are done [9]. So suppose F is a noncompact leaf of y and y has compact leaves. Since ^ has no exceptional minimal sets [9], there is a compact leaf T of ^ such that T is in the closure of F. Let A: be a point of F and cx.{x) a non zero element of 7Ti(F, x). Let X be a vector field on V such that the integral curve of X through x is closed and homotopic to a(A:), and all the integral curves o f X o n F are closed. X is easily constructed using the action 9 (cf. [6]). Since T is in the closure ofF, we know the integral curves of X on T are also closed. Now T is an isolated compact leaf at least on one side in V, the side where F intersects a transverse arc infinitely often. Let U be a neighborhood of T, on this side, such that all the leaves of ^ in U, except T, have zero holonomy. Then U contains closed integral curves of X which are on F, so such an integral curve G has zero holonomy. Since G is conjugate to a(;v), it follows that QL{x) has zero holonomy $ thus F as well.

( 1 . 2 ) Suppose (N is not empty and 9 has no compact orbits in the interior of V. Let T be a compact orbit of 9; T c ^V. The leaves which contain T in their closure are homeomorphic to ^k^'s^n-k-l where k==the rank of the kernel of the holonomy map on T.

Proof. — Let F be an open leaf whose closure contains T; F^'P'xR^"1. Suppose 7} is the kernel of the holonomy homomorphism on T. Let T^ be a A-torus embedded in T which lifts onto nearby leaves by the holonomy. Since FDT, we can lift T^ to a A-torus T^ in F. Also ^ : TT^T) -> ^(V) is injective, where i : T<-^V (cf. [4]), hence ^i(T^) embeds in ^(F) and k<_j.

Next we show j<_k. Let xeV and ae^(F, x), a+o. Let X be a vector field tangent to the orbits of 9, such that the integral curves of X on F are closed and the integral curve o f X through x is homotopic to a. Since FDT, all the integral curves

MANIFOLDS WHICH ADMIT R" ACTIONS 247 of X on T are closed. Let C be an integral curve of X on T. We know that G lifts to a closed curve on F, so by ( i . i), the holonomy of G is trivial; i.e. G is in the kernel of the holonomy homomorphism. Hence j^k.

2. The transverse torus and vector field.

Throughout this section, we suppose <p acts on V so that there are no compact orbits in the interior of V and T is a compact orbit in 8V. Let k denote the rank of the kernel of the holonomy map associated to T; k varies between o and n—2. Let YI, . . ., y^-i ^e linearly independent commuting vector fields on V satisfying:

(i) they are tangent to the cp-orbits;

(ii) their integral curves are closed and of period one on T; and

(iii) the integral curves ofY^, .. ., Y^ represent the kernel of the holonomy map on T.

We shall construct an {n—i)-torus T ' c l n t V such that T u T ' bound a trivial cobordism in V, and Y^-i ^ls transverse to T' at each point.

By ( i . i), we know the orbits of Y^+i, .. ., V^-i on T mduce germs in Diffc^R4') which are contractions or expansions, via the holonomy. Here Diffo(R4') is the set of G^germs of diffeomorphisms of 'R.+ to itself, which leave o fixed. After reversing the sign ofY, if necessary, we shall assume the germs are all contractions, for k-}-i<_j<_n—-1.

Choose a metric on V and let Ug be a geodesic collar neighborhood of T isometric to Tⁿ~^lx[p, s], with the obvious product metric. Clearly, if e is small enough, the geodesies normal to T in Ug will be transverse to the orbits of 9. Let^/p1 be the holonomy diffeomorphism associated to the Y^ orbit through x; f^ is the identity for i^i^k and a contraction for k<i<n.

Proposition (2.1). — There is an [n—i)-torus T' contained in Ug such that V^-i ^u transverse to T' and TuT' bound a trivial cobordism.

In an earlier version of this paper we gave a proof of (2.1) which used calculus.

Charles Pugh pointed out to us how one can use a theorem of W. Wilson on the existence of Liapounov functions for uniform stable attractors of vector fields [13]. We present this proof here and in an appendix we give our original proof.

We need some definitions before stating Wilson's theorem. Let X be a vector field on V and let A be a closed invariant subset of V (here V is a compact manifold).

A is called a uniform stable attractor of X if the following conditions are satisfied:

a) there exists an increasing function 8 sending R4' into itself such that rf(X(A ^), A)<s

whenever ^,A)<8(£) and ^ > o ;

248 G . C H A T E L E T A N D H . R O S E N B E R G

b) there exists a neighborhood U o f A such that co(^)cA whenever peU (o)Q&) is the co-limit set of^)$

^ let D(A) be the set ofp such that <o(^)cA; D(A) is an open set, called the basin of attraction of A.

Wilson has proved [13] that if A is a uniform stable attractor for X then there exists a G°° Liapounov function, i.e.

a) there is a C00 function /:D(A)->R+ with /-l(o)=A; and b) X(/)(^)<o whenever /(^)=t=o.

Hence / has no singularities outside A and all the level surfaces of/ are diffeo- morphic. Before proving (2.1) we need three lemmas.

Lemma (2.2). (Action box lemma.) There exists a unique mapping

F r J ^ x l o . ^ U . c V

(where J==[—i, 2]) satisfying the following conditions:

a) F is a ff-immersion^

b) F sends the horizontal plaques Jⁿ~^lx{z} into the leaves ofy\

c) F sends vertical arcs {A}x[o, e] onto the geodesic arcs normal to T;

d) F', when restricted to .P^X^}, is the restriction of the natural covering map: R""¹-^

induced by 9, which sends the i-direction line onto the Y, circular orbit \

e) let XQ^T; then F sends {o}x [o, s] isometrically onto the geodesic arcs issued from Xo, normal to T and pointing inside T.

Proof. — Define first F via e) and d). F obviously extends to Jn~lX [o, s] using b) and c).

a) is clear, for geodesic arcs are normal to y in Ug. Note that each Y^ orbit on T is covered three times by F.

Lemma (2.3). (Commuting contraction lemma.)

Iff^ andf^ are commuting embeddings [o, s]->[o, oo[ andf^ is a contraction towards o, then there exists a K so large that f^f^ is a contraction to o.

Proof. —j^ commuting with/j^, f^f^ is an embedding without fixed point or is the identity (N. KoppePs Thesis). For sufficiently large A, f^f^ is not the identity.

Hence f^f^ is a contraction or an expansion. For fif^[o, e] ==f^fi[p, s], and K may be chosen so large that J^/iE0) s] C o, £- . f^ is therefore a contraction.

MANIFOLDS WHICH ADMIT W1 ACTIONS 249

3, Attraction Lemma.

There exists s and S>o such that whenever X is a G¹ vector field on R"""¹ and |X|co<8,

(

then Y==<D» — — — + X ) generates a flow having T as a uniform and stable attractor^ Ug being

^n-l /

in the basin of attraction of T.

— Look at the application F of Lemma (2.2) (action box lemma). If Y==<lJ—-+x),

V^n-l /

F*Y is a G¹ vector field defined on Jⁿ~^lx[p, s] (F is a G^immersion); F*Y has no vertical component and may be chosen arbitrarily close to ——— for a suitable choice of S.

a

^n-l

Let I==[o, i], AQ==ln~2x{o}x[o, s], A l = = In~2x { I } x [ o , s] and xe\. I being interior to J, choose 8 such that the positive orbit of F*Y through x crosses A^ before reaching the boundary of ^"^[o, e]. Let x be the point of intersection of \ with the orbit. Via F, x is identified with a point X^AQ and hence may be written in the form (Xi, . . ., X^_i, o, ^) where

^=/KK^+^•••o/KK^+/o•••o/n-l^) ^ ^ = ( X , , . . . , X , _ ^ 0 ^ ) .

Recall that for I ^ J ^ ^ — K + I , /g+j are t^ contracting holonomy diffeomorphisms associated to the circular Y^+j orbits.

Using the contraction commuting lemma, we choose N such that

./K+l° • • • °fn-2°fn-l

is a contraction. For s and 8 small, we may build a sequence (^, x, ^, . . ., ^-i? ^N-I? ^N) where the F*Y orbit through x^ crosses A^ at ^ and ^ being identified via F with x^^

in AQ. So if x=={\, . . ., X^_i, o, 2'), then x^==(\[\ .. ., X^_i, o, h{z)) where

A^^n/^o/^,^).

Thus we have shown that the vertical coordinate of any Y-orbit tends to o in a manner dominated by a fixed contraction f^~^o.. .0/^1) as we proceed along the orbit in forward times, i.e. T is a uniformly stable attractor.

Let us prove now Proposition (2.1).

— The choice of the YJs on T allow us to write T as a trivial fibration SxS^

where S is a manifold diffeomorphic to T^'^nd transversal to the circular orbits ofY^_^

which are the fibers of that fibration. Over these circles, consider the normal geodesic fibers o f U g . This gives a two dimensional foliation of Ug by cylinders. Gall it ^$

^ is clearly transversal to e^.

249

250 G . G H A T E L E T A N D H . R O S E N B E R G

Let Y ^ _ i = = X + Y where Y is tangent to e^n^ and orthogonal to X; clearly Y'n-iW—YM ==X(^) tends to o when 6f(^, T) tends to o. Due to the attraction lemma, Y admits T as a uniform stable attractor. Let V^ = Ug, V^== V^ and let (3 be a bump

3 3"

function such that (B==i on V^ and (B=o outside Vg. Let Z==(BY+(i—(B)Y^_i.

It is easy to check that Z admits T as a uniform stable attractor and hence there exists a Liapounovfonction/forZ. For £ > £ o > — , Z=Y^_^ and/"^So) is transversal to Y^_i.2s

£

For ->£i>o, /^(si) is transverse to Y;/"1^) is diffeomorphic to/"1^). It remains o

to prove/"^i) is a (%—i)-dimensional torus for/"1^) will be then a torus satisfying conditions of (2.1).

Y being transverse to/"1^),/"1^) is transverse to ^. Let ^ be the leaf of ^ through x; j^n/'"1^) is a compact one-dimensional manifold and hence diffeomorphic to a circle. Writing T=2xSi and x={\,s) here Xe2 and seS^y one produces a family of embeddings of S^, (^)^gs such that ^(S^^n/-1^). We define now an application TT : SxS^ -^/-l(£l) by 7c(X, ^)=7r^(J) which is clearly an embedding.

Proposition (2.1) is thereby proved for S is diffeomorphic to T⁷¹"².

Proof of Theorem 1. — We now assume 8V is not empty and 9 has no compact orbits in the interior ofV. Let T, T', and Yi, . . ., Y^ be as in section 2; so that Y^_^ is transverse to T' and pointing into V along T', i.e. Y^-i points out of the tubular neigh- borhood of T. Let F be an orbit of 9 which intersects T' and let L be a connected component of FnT'.

Lemma ( 3 . 1 ) . - ^Y,_^,L)=F.

Proof. — We know F is diffeomorphic to T^xR"'"^""¹ (in the leaf topology) and we have a covering map n : R"""1-^ induced by 9. Since Y^, . .., V^-i define the action 9, we can take ^(Y^i) = - — — where (^, . . ., ^_i) denote the usual coordinates

a

^n-l

in R""1. Let X denote ,——, and let W be a connected component of TT'^L). It

^n-l

suffices to prove that each orbit of X starting at a point of the hyperplane x^_^==o, intersects W, since this implies UX(^, W)==Rn~l.

Now W is a closed submanifold ofR^^ of codimension one, and X is transverse to W, and makes an angle with W that is strictly bounded away from zero, since Y^-i is transverse to T'. Clearly, the set of points of the hyperplane ^_i = o, whose X orbits intersect W, is an open non empty set 0.. It suffices to show Q is closed. Let zeO., and ^e0, satisfying: lim ^==2' and for each n, there exists ^eR, such that X(^,^)eW.

If some subsequence of (Q converges to a number t then we have X(^, -2:)eW; hence we can suppose no subsequence converges. Let (^) be a subsequence of (Q such

MANIFOLDS WHICH ADMIT R^ ACTIONS 251

that | ^ — ^ + i | ^ i and | ^ n — ^ n + i l ^ - • Let E(%) denote the line segment joining ^

72

to ^i and consider (E(^)xR) nW. This is a curve in W with endpoints X(S^, z^) and X(S^i, ^n+i)- There exists a point U^ on this curve where the tangent to the curve is parallel to the cord joining the endpoints. The angle this cord makes with X tends to zero as n->co, which contradicts the fact that the angle between X and W is strictly positive.

Lemma (3.2). — Let F, W, L, T and T' be as in (3.1). Then there exists a compact orbit TI 0/9 such that FDT^ and T^+T.

Proof. — Let WQ==W and W^=-=X(^,Wo) for each positive integer n. By an argument analogous to that of (3.1), one sees that the distances ^(W^.W^J tend to infinity as s—>co. Let L()==L and L =Y ^(%, L()), so that lim ^(L/,, L , , , ) = o o ,

S —> 00

where the metric is that induced by n. We define Q = D E^, where E^ is the connected

n

component of F—Ly^ towards which Y^_i points on Ly^. Q. is an intersection of a nested family of compact sets, hence £1 is not empty and compact. We claim 0. is invariant under the 9 action: clearly n=={j^eV[ there exists x^eE^ and x^->y}. Let F(j^) be the orbit of 9 by y^Q and let _/eF(^). Let [jsj'] denote a path in F(j^) joining^ to y and let [^, x^] be the holonomy lifting of this path to the leaf of x^. By construction we have d{x^y x^) bounded above by some number /', independent of n. Since

^(Ln,L^)-^00

as s->co, we can choose a subsequence of %), call it (j^J, such that j^eE^. Thus y^O and ^ is invariant. Thus Q contains a 9-minimal set, which must be a compact orbit by Sacksteder's theorem. Since V^-i pomts away from T, this compact leaf T^cfi, is different from T.

(3.3) Let Vⁿ be of rank n—i and let 9 be an action qfK¹"¹ on V such that the only compact orbits of 9 are in 8V and 8V is not empty. Then V is homeomorphic to ^"^xL

Proof. — We use the notation of (3.1) and (3.2). From these lemmas, it follows that the open leaves having T int heir closure are homeomorphic to the open leaves having T^ in their closure, i.e. to ^kx1Sn~k~l, where k is the rank of the kernel of the holonomy map of T. Now since all the integral curves of Y^, . . ., Y^; are closed in F, and FDT^, we know they are also closed in T^; hence the A-tori in T^ spanned by the orbits of Y^, . . ., Y^; represents the kernel of the holonomy map of T^. Now the orbits ofY^+i, . . ., Y^_i are not necessarily closed but we can choose vector fields (from lines through the origin in R^1) Vjc+D ' ' •? ^n-i? ^^ lhat Y^, . . ., Y^;, Y^i, . . ., V^-i are linearly independent, commute, are tangent to the 9 orbits, and all the integral curves of Y^i, . . .3 V^-i m ^i are closed. Clearly this can be done so that Y^-i ls

251

252 G . C H A T E L E T A N D H . R O S E N B E R G

G°-close to Y^_i. We choose V^-i so ^^ Aat V^-i ls a^so transverse to T'. Now we go through the construction of a torus T[, bounding a collar neighborhood with T^, such that y^-ils transverse to T[; this is (2. i). Letting Y denote Y^_^, we now have a linear vector field Y transverse to both tori T' and T[. We know the set of points A in T' whose Y-integral curve intersects T[ is an open non empty set. By the same reasoning, the complement of A in T' is open; hence A==T'. Now using the integral curves ofY, it is easy to construct a homeomorphism between V and ^ⁿ~^lxI.

Proof of Theorem 1. — The proof follows from (3.3)3 and a reasoning identical to that on page 462 of [7].

Remarks 1. — A basic question remains unanswered: suppose <p is a locally free action of R"""1 on a closed manifold V", with no compact orbits. Then we know V^ fibres over a torus with fibre a torus, hence V^ fibres over S1 with fibre F (this also follows from [10]). Is F homeomorphic to Tp"1?

2. Suppose V^ is a closed, orientable, bundle over S1 with fibre M. Then there exists a diffeomorphism f : M—^M such that V is obtained from M X I by identifying points (/M, i) with {x, o) for xeM. We claim that if /* : H^M, R) -> H^M, R) does not have one as an eigenvalue, then every locally free action ofR^"1 on V has a compact orbit. To see this, first observe that f* does not have i as an eigenvalue if and only if rank H^V, R) == i [i i]. Now suppose SF is any foliation ofV of codimension one, class G2 and with no compact leaves. By [12], we can suppose L is a covering space of M for L a leaf of <^. We have an exact sequence of free abelian groups:

0 -^(F) /7T,(L) ->^(V) /7T,(L) ->^ ->0.

Since H^V.R^R, the last two groups are of rank one. Hence n^(L)=n^'F) and L must be compact.

APPENDIX

Proof of (2.1)

Notation. — If X is a vector field on V, t\->'X.{t, x) will denote the integral curve of X passing through x at t==o. For AcV, X(^, A)=={X(^, x) \xeA}, and

X(^&],A)=^X^A).

If A:eT, we define a^)=Y,([o, i], x) for z = i , . . . , ^ — i , and T^x) is the z-torus in T which is the orbit through x of the Reaction determined by Y^, . . . , Y^.

If x is on the normal arc through xeT and if the holonomy germs are defined on x,

MANIFOLDS WHICH ADMIT R" ACTIONS 253 then we denote by T^(^) the lifting of Tjc{x) into the leaf of ~x^ given by the holonomy.

Let N be a unit vector field on V, normal to the orbits of <p and pointing into V along T (with respect to some metric on V). Let U==N(I, T) where I=[o, i]. We may suppose U is a tubular neighborhood of T in which the holonomy liftings of a^A:), ..., v-n-iW are defined, for xeT. Let f^ be the holonomy diffeomorphism of

^iM; i<z<_n—k—i.

Let TT : U->T be the projection along N orbits. If xeT and xen'1^), let a^(^) denote the holonomy lifting ofa^) starting at x; for i^i^k, ^{x) is an embedded circle, and for i>k, oi^x) is diffeomorphic to I. For xeT and for all xen'1^), the a^(^) form a one dimensional foliation of U. Let G( be a vector field in U, tangent to this foliation, and coinciding with Y( on T.

We fix a base point A^eT and we let 0^=0^0), T^=T^), etc., and define A,=N(I,TJ.

Let E^(A.) be the vector bundle of exterior products of order t of vectors tangent to A.. We identify E^(A-) with A^xAR-^; so sections of E^(Aj) are functions from A^.

t

to AR9. We give these sections the canonical norm.

Let / be a function defined in a neighborhood of o such that lim/(A:) = o. We write f==^{x) if

a;->0"

f{x)==ax+x^(x),

with ^4=0 and S[x)->o when x->o. Finally, we let (B^==Y^A . . . AY^_i.

Proposition ( 2 . 1 ) . — For each j , i ^ J ^ ^ — k — i , there is a family of tori G(A+j), satisfying:

c^) there is a neighborhood Uj of Tj^^j and the G{k+jYs are a foliation of Uy by tori of dimension k +j;

c^) there is a section g^^^ of E^.(A^) such that gjc+j{x) represents the tangent space at x to G{k+j){x) and

(&+,AP^,)^+0

for all peV^T^

c^) on T, G^,(A;)==T^,.

Remark. — In particular c^) implies

gn-l^n-l^0 m U^-T.

Hence there exist {n—i)-tori, transverse to Y^_^, as close to T as we wish.

Proof of (2.1). — We proceed by induction onj; firstc c cylinders '9 are constructed in Txl, k-\-i<_j<^n—2, and then these cylinders are closed, to give tori, by the map F. defined by the holonomy of a,.

We start by constructing the foliation G(k+i). Let Ui^Tc^T^i), and

254 G . C H A T E L E T A N D H . R O S E N B E R G

let (6, z, X) be coordinates for T^xIxJ where 6 ==(6^ . . . , Qj,)eT\ and I=J=[o, i].

Let Pi iT^xIxJ-^Ui be defined by:

Fi(o,o,o)=^o Fi(o, ^ o)=N(^o)

Fi(6, 2', X) is the endpoint of the holonomy lifting of the arc in T given by:

^,Y(X^F,(e,o,o)), o<_t<_i, starting at F^(6, z, o)==N(^, Fi(6, 0,0)).

Here we have identified ^JC==J^k|Zk with T^ by the linear diffeomorphism ( o , . . . , 6 , , . . . , o ) h > Y , ( 8 , ) ( ^ ) .

By definition of F^ we have:

(

— DFi — ) is colinear with G. for I^J^A,

ao,/

— F^ sends the tori TA ;x{2'}x{x} to the holonomy liftings of the tori T^(Fi(o, o, X)) to the point F^o, z, X),

==^ck+l==^yk+l ^{on T}?

— F^ send the segments { 6 } x l x { x } to the orbits of N starting at Fi(6, o, X),

— the segments {9}x{^}xJ are sent to oc^i(Fi(6, z, o)),

— FI is a local diffeomorphism to U^.

From these remarks, it is easy to see that the map 2'i-^F^(6, z, X) (respectively Xh->F^(6, ^, X)) is a reparimetrization of the N-orbits (orbits of C^i). Hence there exist functions (p^ and ^, invertible in z and X such that

/ a \ a<pi

D^(-}=-tlN

^z] 8z

w l ^ } ^c

DFl[^]='^ck+l

(both 9^ and ^ have strictly positive derivatives on T^xIxJ).

Now we construct a family of curves, y^S? ;2')? m T ^ x I x J Yi(6,^) : X ^ ( Z i ( 6 , Z , X ) , X )

satisfying conditions A) and J?J:

A) For fixed 6, z, Fi(yi(6, 2:)) is a closed curve in U^, of class G¹. For 6 and z in a neighborhood of T^x^}, the F^(Yi(6, Z)) form a one dimensional foliation of a neighborhood of T^^^ in A^i,

MANIFOLDS WHICH ADMIT R71 ACTIONS ₂₅₅

B) Let A^e.O^-/^), where ^=Fi(6,o,o), and let A^)==Ai(o, ^). Then we require that:

t-,(.,)

(here A^ is the function ^t-^A^)). The condition B) is not necessary to construct G(^+i); however, it is necessary to insure the transversality relation c^) when we construct G{k-{-j), J>i.

Lemma (2.2). — There exists in T^xIxJ, a family of curves Yi(6, ;?), satisfying conditions A) and B).

Proof of (2.2). — Let y' be the tangent vector field to the y^ curves, with the X-parametrization. Then condition A) can be written:

W (DF^A(DF,)^=O

where b==(Q,f^(z), i), A:=Fi(e, o, o), and a = ( 6 , ^ o ) (cf. figure i).

Fi(d)

7t(a) X(b)

FIG.

An easy calculation shows that (i) can be written:

az.

- =K(6,.)-^

ax

where K is a strictly positive function. Therefore, we can rewrite A) and B) as:

az,

"ax

az, 1^

f=^,).

A tedious, but simple calculation, shows that the cubics (see fig. i):

i+Koj ^ 1

X^,Z,==A,(e,.) ^-al^+ls-

\ LV'+^O / \ K-o+•+3-^^-^ W1

256 G. C H A T E L E T A N D H. R O S E N B E R G

satisfy these equations, where

Ko=sup K(6, z), (6, z)e^kxlx{l}.

Now, one can write:

aZ^

~8\ =A,(6,^(9,^,X),

where g>o on T^xIxJ. Also —>o on T ^ x ^ A J x J for a suitable A, o<A<i.

az.

B^

Hence the curves yi form a foliation of T^x^, A]xJ. Their image by F^ is a foliation (of class G1) of a neighborhood V^cU^ of Tjc+i m \+i- This completes the proof of (2.2) (see fig. 2).

FIG. 2

We can now define G(^+i). The submanifolds:

H(^U^(e,.)x{e}

are diffeomorphic to Tkx [o, i] and form a foliation of ^JCX [o, AJxJ. Hence their image by F^ is a foliation of V^ by tori G(k +1) (figure 2). We now check conditions c^) and c^).

We have a G2 vector field y' i^ T^x [o, A]xJ; y' ls ^e tangent field to the YI curves with the X-parametrization. This field induces a natural action of R on the exterior products of vector fields, which we note by F(X): F(X) is the differential of the map induced by y' of T^x [o, h~\ x{o} to T^kx [o, h] x{X}. Then the tangent space to H(-2:) at the point (6, z, X) is given by:

/ a a \

rw ^ A . . . A , . AY(^.X).

\^i ^/(e,z,o)

MANIFOLDS WHICH ADMIT R" ACTIONS

^

257

Now F, sends the tori T1ex{z}x{o} to the trivial holonomy liftings of the T • hence the tangent space to G(A+i) at p=~F^9, z, X) is given by:

(&^=(DF,or(X)^A.. .A^ADF,OY')^

[^i " • • • " ^ ' ^

"

^

• ^

^^^a^

\ i k/

We recall that:

-•(^IN, ^)^c. ••(.zh^. ^,-^c,,,.

Let S'(Ai) denote a function such that — ^ tends towards a limit a- thenA^ ————> «. —x... ^, 3'(Ai)=aAi+o(Ai),

(with a not necessarily different from o and a(^)=^s(\), s(Ai)^o whenever Ai-^o). We take the tangent spaces to the trivial holonomy liftings of the T^, to be given by a section of E^(A^), equal to Y^A ... ^ on T^.

Let G^i(X) denote the action induced by G^ on the vectors tangent to \ (if Y is tangent to the y-orbits, then so is C^(X)(Y)). Then we obtain for g^:

(&+i)y=(C^i(X)o(^)^^^+3-(A^)Op)A(C^i+^(Ai)N)p,

where p='F^9, z, X) and 0 is a section ofE^(A^) defined on V^. We can rewrite this as:

c!&+lAC^l(X)o^+^(A,)C^l(X)o^AN+(T(A^)G^^An+<T(Al)NAO.

Now ^k+i^^+i^+iW'^gk is zero, since it is a linear combination of exterior products of n vectors tangent to the y-orbits. Hence:

&+lAPA+l=^(Al)C^^(X)o^ANAp^^+(7(Al)C^+lAQAp^^+(T(Al)NA^Ap^l.

We have:

Q+lW°&ANAp^i =Y,A . . . AY^NAY^A . . . AY^.i

on T^i. Hence for all points p in a neighborhood V^ of T^+i, we have:

IQ+iW&ANA(3^|p>a>o.

We want to show (&+iA(3^),,+o, forj&in a suitable neighborhood of T , , in A, , Dividing by (TI (A,) (+o if z+o):

Sk +1 ^ Pfc 4-1

-^J-=P.+C,^AQ'Ap,^+^(Ai)NA^Ap,^,

where |pjp>a>o in Vg, and Q' is a bounded section on Vg-T^,, ^(Ai)->o as Ai->o. Since G^i=Y^i on T^^, the second term is less than a/g if^ is in some

257

258 G . C H A T E L E T A N D H . R O S E N B E R G

neighborhood V^ of Tjc+f Also |<?(Ai) | [ N A n A [ B ^ J < a / 3 if p is in some neighbor- hood V4 ofT^i. Hence for peV^nV^nV^,

Ig^lAp |^

^i(Ai) 'j

which proves the theorem for ^==1.

It is useful for the induction to write ^_^ in the form:

& + i = a ^ i + a ( A i ) ^ "

where a^_^ is a section of E^_^(A^^), defined in a neighborhood of T^.^ and equal to Y i A . . . A Y ^ Y ^ i on T^i.

Construction of G ( A + J + i ) . — L e t A^^)^.?—/^4'^), where j-^o denotes the normal N-coordinate.

Fondamental little lemma:

lim , ^{K + j} m^j.

-OAK+,+I

Following (i), one may find an homeomorphism H : [o, s] -> [o, s'] such that K ' ^ K + j K ^ ^ K + j ^and H ~ "^l/ K + j + l H = X K + j + l where XK+^ and ^K+J+I ^{are the} homo- theties the ratio of which are X^^j and XK^^^ (recall ^+1 ^-^fK+j+i are contractions).

Define on [o, s] a metric S such that S(A:, x')== \tl~l[x)—'H.~l{x/)\ — this metric is topologically equivalent to the classical one — and S{x, o)—^o whenever x->o. We prove ——3 K + J has a limit when x->o (with respect to 8); we shall then be

^^UK+j+lW)

over. Then let /=/K+^ S=fK+3+^

S^/^JH-^-H^)!^ \H-l(x)-\^oH-l(x)\

Ux,g{x)) IH-^^-H-^)! \H-\x)-^^oH-\x)\

^ /(.))_ I-XK,,

^S^^-i-XK,,,^'1 w

when ^-^o, 8(A<,o)->o and p->———^^-±1-,

I—XK 4 - j + l

Our inductive hypothesis asserts the existence of the foliation G{k-{-j) and a section gj^^y satisfying:

&+j=a^.+CT(A^)Q,

where a^ is a section of E^(A^), defined in a neighborhood U of Tjc+j m ^fc+j?

which is a linear combination of vectors tangent to the (p-orbits, and equal to Y^A . . . AY^.

on T^. Q, is a section of E^(A^.) defined in U. Henceforth, we work in U.

MANIFOLDS WHICH ADMIT R72 ACTIONS 359

Since the G{k+j) form a fbliation of U transverse to the normals, we can construct, by the holonomy, a map F^ satisfying:

F^i sends Tk+jxIxJ to U and is of maximal rank;

F,+i(o, ^,o)=N(^o);

F^ sends the tori T^x^X^} to the tori G{k+j) passing by F^ 1(0, z, o);

restricted to T^'x^xJ, we have:

DF - ⁺¹ (^) ^=Y/ ' ^I ^ ^< ^

"^(a^^"^

(

D F j + l^A• "A^ )= = & + j i n A f c + j ;

F^i is the holonomy lifting, restricted to the plaques {6}xlxj, i.e. ( F = = F . , i ) F(6, z, X) is the endpoint of the holonomy lifting of the path Y^..^([o, X], F(6, o, o)) to the point F(9, z, o).

Exactly as the case j'=i, we have a family of curves Yj+i(9, z) satisfying the conditions A) and B ) , with F^ replaced by F==F^i. This gives us a foliation of Tk+jxIxJ by submanifolds diffeomorphic to Tk+jxI, and closing the cylinders by F we obtain a foliation by tori G(^+^'+i). We must verify c^).

Let g=g^^^ G=C^.+i, A==A^^i and G⁺=C^^l. Then we have:

5-(C+^(A)N)A(e(X)o^^.+<T(A)^)

=GAG*(X)o&+,+(T(A)NA^'+^(A)NAG*(X)o&+,+a(A)GAQ'.

Now we write ^.==a^.+(7(A^)^ (a section defined in A^.), to obtain (defined in \+^^):

^=CAG*(X)oa^,+(CAG*(X)a)(T(A,^)+(?l(A)NAG+(X)oa,^

+CT(A)a(A^^.)^+G(A)GAQ'+(7(A)NAa".

Notice that CAG*(X)oa^^^. is a linear combination of products of order k-\-j-{-i of vectors tangent to the leaves, and on T^..^, it equals Y^^iAY^A. . . AY^..

Composing ^ with (B^.+i:

— the term CAG*(X)a^+^A^^.^i=o since it is a multiple of n vectors tangent to the 9 orbits;

- N A C - ( X ) o a ^ , = = N A Y , A . . . A Y ^ , onT^,^.

Dividing by ^(A) we obtain:

^^-^NAC-Ma^.+^^GAG-MQ+^A)^.

260 G . C H A T E L E T A N D H . R O S E N B E R G

Now, as j->o, A^./A is bounded hence (T(A^)/ai(A) is bounded. Q." is a bounded section of ^^j+i^+j+i) ^d <?(A) ->o as A-^o. Hence, in a small enough neighborhood ofT^..^, we have ^AJB^^+O, which completes the proof of (2.1).

BIBLIOGRAPHY

[i] G. GHATELET, H. ROSENBERG, Un theoreme de conjugaison des feuilletages, Ann. Inst. Fourier, 21 (1971), fasc. 3, 95-io6.

[2] G. CHATELET, H. ROSENBERG, D. WEIL, A classification of the topological types of Reactions on closed orientable 3-manifolds, Publ. math. I.H.E.S., 43 (1974), 261-272.

[3] G. JOUBERT et R. Moussu, Feuilletage sans holonomie d'une variete fermee, C. R. Acad. Sci., 270 (1970), A-5o7-509.

[4] S. P. NOVIKOV, Topology of foliations, Trudy Mosk. Math. Obsch., 14 (1965), 248-278 (== Trans. Mosc. Math.

Soc., 14 (1965), 268-305, transl. J. Zilber).

[5] H. ROSENBERG, Foliations by planes. Topology, 7 (1968), 131-141.

[6] H. ROSENBERG, The rank of S^S^ Amer. Journ. of Math., 87 (i965), 11-24.

[7] H. ROSENBERG, R. ROUSSARIE and D. WEIL, A classification of closed orientable 3-manifolds of rank two, Ann.

of Math., 91 (1970), 449-464.

[8] R. Moussu et R. ROUSSARIE, Relations de conjugaison et de cobordisme entre certains feuilletages, Publ. math. I.H.E.S., 43 (1974), 144-167.

[9] R. SACKSTEDER, Foliations and pseudo-groups, Amer. Journ. of Math., 87 (1966), 79-102.

[10] D. TISCHLER, On fibering certain foliated manifolds over S1, Topology, 9 (1970), 153-154.

[n] J. PLANTE, Anosov flows (to appear).

[12] R. Moussu, Feuilletage sans holonomie d'une variete fermee, C. R. Acad. Sci., t. 270 (1970), A-i3o8-i3ii.

[13] F. W. WILSON, Smoothing derivatives of fonctions and applications, Trans., Amer. Math. Soc., 139 (1969), 413-428.

Manuscrit re§u Ie 14 juin 1973.